Binomial Theorem

If the sum of the co-efficients of all the positive even powers of $x$ in the binomial expansion of $\left(2x^3+\dfrac3x\right)^{10}$ is $5^{10}-\beta\cdot3^9$, then $\beta$ is equal to ______.

The remainder when $(2021)^{2023}$ is divided by 7 is :

If $\left({}^{40}C_0\right)+\left({}^{41}C_1\right)+\left({}^{42}C_2\right)+\dots+\left({}^{60}C_{20}\right)=\dfrac mn\,{}^{60}C_{20}$, $m$ and $n$ are coprime, then $m+n$ is equal to ______.

If the coefficient of $x^{10}$ in the binomial expansion of $\left(\dfrac{\sqrt x}{5^{1/4}}+\dfrac{\sqrt5}{x^{1/3}}\right)^{60}$ is $5^k\cdot l$, where $l,k\in\mathbb N$ and $l$ is co-prime to 5, then $k$ is equal to ______.

Let $S=2+\dfrac67+\dfrac{12}{7^2}+\dfrac{20}{7^3}+\dfrac{30}{7^4}+\dots$. Then $4S$ is equal to

If the sum of the coefficients of all the positive powers of $x$, in the Binomial expansion of $\left(x^n+\dfrac2{x^5}\right)^7$ is 939, then the sum of all the possible integral values of $n$ is ______.

If $\displaystyle\sum_{k=1}^{31}\left({}^{31}C_k\right)\left({}^{31}C_{k-1}\right)-\sum_{k=1}^{30}\left({}^{30}C_k\right)\left({}^{30}C_{k-1}\right)=\frac{\alpha(60!)}{(30!)(31!)}$, where $\alpha\in\mathbb R$, then the value of $16\alpha$ is equal to

The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2x^3+\dfrac3{x^k}\right)^{12}$, $x\ne0$ is $2^8\cdot l$, where $l$ is an odd integer, is ______.

The term independent of $x$ in the expansion of $\left(1-x^2+3x^3\right)\left(\dfrac52x^3-\dfrac1{5x^2}\right)^{11}$, $x\ne0$ is :

If the constant term in the expansion of $\left(3x^3-2x^2+\dfrac5{x^5}\right)^{10}$ is $2^k\cdot l$, where $l$ is an odd integer, then the value of $k$ is equal to :

Let $A=\begin{pmatrix}2&-1\\0&2\end{pmatrix}$. If $B=I-{}^5C_1(\operatorname{adj}A)+{}^5C_2(\operatorname{adj}A)^2-\dots-{}^5C_5(\operatorname{adj}A)^5$, then the sum of all elements of the matrix B is

Let $n\ge5$ be an integer. If $9^n-8n-1=64\alpha$ and $6^n-5n-1=25\beta$, then $\alpha-\beta$ is equal to

Let the coefficients of $x^{-1}$ and $x^{-3}$ in the expansion of $\left(2x^{\frac15}-\dfrac1{x^{\frac15}}\right)^{15}$, $x>0$, be $m$ and $n$ respectively. If $r$ is a positive integer such that $mn^2={}^{15}C_r\cdot2^r$, then the value of $r$ is equal to ______.

For two positive real numbers a and b such that $\dfrac1{a^2}+\dfrac1{b^3}=4$, the minimum value of the constant term in the expansion of $\left(ax^{\frac18}+bx^{-\frac1{12}}\right)^{10}$ is :

If the maximum value of the term independent of $t$ in the expansion of $\left(t^2x^{\frac15}+\dfrac{(1-x)^{\frac1{10}}}t\right)^{15}$, $x\ge0$, is K, then 8 K is equal to ______.

The remainder when $(11)^{1011}+(1011)^{11}$ is divided by 9 is

If the coefficients of $x$ and $x^2$ in the expansion of $(1+x)^p(1-x)^q$, $p,q\le15$, are $-3$ and $-5$ respectively, then the coefficient of $x^3$ is equal to ______.

$\displaystyle\sum_{\substack{i,j=0\\i\ne j}}^{n}{}^nC_i\,{}^nC_j$ is equal to

The remainder when $(2021)^{2022}+(2022)^{2021}$ is divided by 7 is

Let for the 9$^{\text{th}}$ term in the binomial expansion of $(3+6x)^n$, in the increasing powers of $6x$, to be the greatest for $x=\dfrac32$, the least value of n is $n_0$. If k is the ratio of the coefficient of $x^6$ to the coefficient of $x^3$, then $k+n_0$ is equal to :